Answer :
Sure! Let's complete and solve each series step-by-step.
### 1. Series: [tex]\[12 + 16 + 20 + 24 + \ldots = \sum_{n=1}^{10} (8 + 4n)\][/tex]
We need to determine the series and then find the sum:
1. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 4 \cdot 1 = 8 + 4 = 12 \)[/tex]
2. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 4 \cdot 2 = 8 + 8 = 16 \)[/tex]
3. For [tex]\( n=3 \)[/tex]: [tex]\( 8 + 4 \cdot 3 = 8 + 12 = 20 \)[/tex]
4. For [tex]\( n=4 \)[/tex]: [tex]\( 8 + 4 \cdot 4 = 8 + 16 = 24 \)[/tex]
5. [tex]\(\vdots\)[/tex]
6. For [tex]\( n=10 \)[/tex]: [tex]\( 8 + 4 \cdot 10 = 8 + 40 = 48 \)[/tex]
So, the sequence is: 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
Now, summing these terms:
[tex]\[ 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48 = 300 \][/tex]
### 2. Series: [tex]\[13 + 17 + 21 + 25 + \ldots + 57 = \sum (5 + 4n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=2 \)[/tex]: [tex]\( 5 + 4 \cdot 2 = 5 + 8 = 13 \)[/tex]
2. For [tex]\( n=3 \)[/tex]: [tex]\( 5 + 4 \cdot 3 = 5 + 12 = 17 \)[/tex]
3. For [tex]\( n=4 \)[/tex]: [tex]\( 5 + 4 \cdot 4 = 5 + 16 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=14 \)[/tex]: [tex]\( 5 + 4 \cdot 14 = 5 + 56 = 61 \)[/tex]
So, the sequence is: 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61.
Now, summing these terms:
[tex]\[ 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 + 53 + 57 + 61 = 481 \][/tex]
### 3. Series: [tex]\[8 + 11 + 14 + 17 + \ldots + 29 = \sum_{n=2}\][/tex]
To find the series and its sum:
1. For [tex]\( n=0 \)[/tex]: [tex]\( 8 + 3 \cdot 0 = 8 + 0 = 8 \)[/tex]
2. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 3 \cdot 1 = 8 + 3 = 11 \)[/tex]
3. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 3 \cdot 2 = 8 + 6 = 14 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=7 \)[/tex]: [tex]\( 8 + 3 \cdot 7 = 8 + 21 = 29 \)[/tex]
So, the sequence is: 8, 11, 14, 17, 20, 23, 26, 29.
Now, summing these terms:
[tex]\[ 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 148 \][/tex]
### 4. Series: [tex]\[12 + 15 + 18 + 21 + \ldots + 39 = \sum (3 + 3n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=4 \)[/tex]: [tex]\( 3 + 3 \cdot 4 = 3 + 12 = 15 \)[/tex]
2. For [tex]\( n=5 \)[/tex]: [tex]\( 3 + 3 \cdot 5 = 3 + 15 = 18 \)[/tex]
3. For [tex]\( n=6 \)[/tex]: [tex]\( 3 + 3 \cdot 6 = 3 + 18 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=11 \)[/tex]: [tex]\( 3 + 3 \cdot 11 = 3 + 33 = 36 \)[/tex]
So, the sequence is: 15, 18, 21, 24, 27, 30, 33, 36.
Now, summing these terms:
[tex]\[ 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 = 204 \][/tex]
Therefore, the completed series and sums are as follows:
1. [tex]\(12, 16, 20, 24, 28, 32, 36, 40, 44, 48 = \sum (8 + 4n) = 300\)[/tex]
2. [tex]\(13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61 = \sum (5 + 4n) = 481\)[/tex]
3. [tex]\(8, 11, 14, 17, 20, 23, 26, 29 = \sum_{n=2} = 148\)[/tex]
4. [tex]\(15, 18, 21, 24, 27, 30, 33, 36 = \sum (3 + 3n) = 204\)[/tex]
### 1. Series: [tex]\[12 + 16 + 20 + 24 + \ldots = \sum_{n=1}^{10} (8 + 4n)\][/tex]
We need to determine the series and then find the sum:
1. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 4 \cdot 1 = 8 + 4 = 12 \)[/tex]
2. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 4 \cdot 2 = 8 + 8 = 16 \)[/tex]
3. For [tex]\( n=3 \)[/tex]: [tex]\( 8 + 4 \cdot 3 = 8 + 12 = 20 \)[/tex]
4. For [tex]\( n=4 \)[/tex]: [tex]\( 8 + 4 \cdot 4 = 8 + 16 = 24 \)[/tex]
5. [tex]\(\vdots\)[/tex]
6. For [tex]\( n=10 \)[/tex]: [tex]\( 8 + 4 \cdot 10 = 8 + 40 = 48 \)[/tex]
So, the sequence is: 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
Now, summing these terms:
[tex]\[ 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48 = 300 \][/tex]
### 2. Series: [tex]\[13 + 17 + 21 + 25 + \ldots + 57 = \sum (5 + 4n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=2 \)[/tex]: [tex]\( 5 + 4 \cdot 2 = 5 + 8 = 13 \)[/tex]
2. For [tex]\( n=3 \)[/tex]: [tex]\( 5 + 4 \cdot 3 = 5 + 12 = 17 \)[/tex]
3. For [tex]\( n=4 \)[/tex]: [tex]\( 5 + 4 \cdot 4 = 5 + 16 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=14 \)[/tex]: [tex]\( 5 + 4 \cdot 14 = 5 + 56 = 61 \)[/tex]
So, the sequence is: 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61.
Now, summing these terms:
[tex]\[ 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 + 53 + 57 + 61 = 481 \][/tex]
### 3. Series: [tex]\[8 + 11 + 14 + 17 + \ldots + 29 = \sum_{n=2}\][/tex]
To find the series and its sum:
1. For [tex]\( n=0 \)[/tex]: [tex]\( 8 + 3 \cdot 0 = 8 + 0 = 8 \)[/tex]
2. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 3 \cdot 1 = 8 + 3 = 11 \)[/tex]
3. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 3 \cdot 2 = 8 + 6 = 14 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=7 \)[/tex]: [tex]\( 8 + 3 \cdot 7 = 8 + 21 = 29 \)[/tex]
So, the sequence is: 8, 11, 14, 17, 20, 23, 26, 29.
Now, summing these terms:
[tex]\[ 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 148 \][/tex]
### 4. Series: [tex]\[12 + 15 + 18 + 21 + \ldots + 39 = \sum (3 + 3n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=4 \)[/tex]: [tex]\( 3 + 3 \cdot 4 = 3 + 12 = 15 \)[/tex]
2. For [tex]\( n=5 \)[/tex]: [tex]\( 3 + 3 \cdot 5 = 3 + 15 = 18 \)[/tex]
3. For [tex]\( n=6 \)[/tex]: [tex]\( 3 + 3 \cdot 6 = 3 + 18 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=11 \)[/tex]: [tex]\( 3 + 3 \cdot 11 = 3 + 33 = 36 \)[/tex]
So, the sequence is: 15, 18, 21, 24, 27, 30, 33, 36.
Now, summing these terms:
[tex]\[ 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 = 204 \][/tex]
Therefore, the completed series and sums are as follows:
1. [tex]\(12, 16, 20, 24, 28, 32, 36, 40, 44, 48 = \sum (8 + 4n) = 300\)[/tex]
2. [tex]\(13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61 = \sum (5 + 4n) = 481\)[/tex]
3. [tex]\(8, 11, 14, 17, 20, 23, 26, 29 = \sum_{n=2} = 148\)[/tex]
4. [tex]\(15, 18, 21, 24, 27, 30, 33, 36 = \sum (3 + 3n) = 204\)[/tex]
